Transformer coupling coefficients

inductance calculation itself has nothing to do with the magnetic core. However the ration of the leakage inductance figure to the OCL does determine the coupling coefficient as can be seen from the equation above. ( my earlier posting was a typo, cleared up here)

I suffered a major brain infarction in what follows:

Replace with:

Notice that when the coils are loosely coupled, m will be small, and using this method will require us to subtract two nearly identical numbers (because the L's are much bigger than m), causing severe numerical cancellation. This will happen when k is some number like .5, not very close to 1. For coils that are tightly coupled, the method works ok.

Hello all,

This is a very nice piece of work Phantom, but:

The Phantom wrote:

The difference between your 2 formulas is N2/N1 vs. SQRT(L2/L1) Thus the validity condition should be: The approximation is acceptable if the 2 coils have approximately the same inductance per turn (A sub l), which is obviously only true if the 2 windings have approximately the same geometry (same length, same cross-section, same number of turns, etc.) (L1=N1^2 x Al)

The coupling factor doesn't enter into it. The approximation is acceptable for every possible coupling value [0...1].

It is very possible that a lot of the confusion around transformers arises from attempts to model unequal A sub l values of primary and secondary coils by tweaking k

Paul

inductance per

approximately the same

every

Are you saying that since "The coupling factor doesn't enter into it.", V2 = (N2/N1) * V1, and k shouldn't appear in this expression?

attempts

inductance per

approximately the same

for every

No, I mean to say that the value of the coupling factor is of no importance in the validity of "V2 = (N2/N1) * k * V1"

Paul

inductance per

approximately the same

for every

the

I see. You're saying no matter what k is, *if* N2/N1 is a good approximation to SQRT(L2/L1) because of the particular geometry of a pair of inductors (...if the 2 windings have approximately the same geometry (same length, same cross-section, same number of turns, etc.)), then V2 = (N2/N1) * k * V1 will give approximately the same result as V2 = SQRT(L2/L1) * k * V1. I would have to agree.

Instead of saying:

"The N2/N1 ratio is an approximati "The N2/N1 ratio is an approximation which is only acceptably accurate, in all cases of inductor geometry, for closely coupled inductors such as found in ordinary transformers (when they *are* closely coupled, as is the usual case)."

This would allow for the possibility that N2/N1 might equal SQRT(L2/L1) for some particular geometry even if k is low.

Consider it said.

I could have also dealt with this particular case by stating my main point in a different way:

The formula V2 = SQRT(L2/L1) * k * V1 *always* gives a correct result, no matter what the geometry of the inductors is, or what k is. The formula V2 = (N2/N1) * k * V1 *doesn't always* give a correct result for all k, and is especially likely to give an incorrect result when k is substantially less than 1.

inductance per

approximately the same

every

attempts

The Asub l is less sensitive to winding geometry if coupling is performed by relatively high permeability material ie not by air.

I think you're both saying the same thing.

RL

inductance per

approximately the same

for every

in the

'Likely' is probably right, since it is _impossible_ for 2 coils which are closely coupled to have very different Al values, but when k is low, Al1 and Al2 _can_ differ substantially. They are subject to the following equations: k SQRT(Al1/Al2)

inductance per

approximately the same

for every

in the

closely coupled

substantially.

This is just another way of saying that when k is low, N2/N1 can differ substantially from SQRT(L2/L1), but when k is nearly 1, N2/N1 scarcely differs from SQRT(L2/L1). That is the main point I've been making in the larger thread. Do you see some particular usefulness in restating it in terms of Al1, Al2?

same inductance per

approximately the same

acceptable for every

in the

closely coupled

substantially.

thread.

It helped me understanding the difference between the two. I feel very comfortable when geometric differences between the coils are reflected by a parameter. I had some bad trouble simulating low-k transformers (a very effective way of looking at electromotors) before I realised this.

I think my original point was lost. You're right of course when you say: all you ever have to know about a transformer is k, L1 and L2. If you're happy with an approximation and you know k to be approximately 1, then you can safely approximate L2 with L1 x(N2/N1)^2. What I was commenting on was: "only acceptably accurate for closely coupled inductors". I'm saying: even with low coupling, the approximation _can_ still be valid, just imagine 2 identical air-coils at considerable distance. The approximation is also valid for the coils of a 3-phase motor. There, the coupling is also significantly smaller than 1.

Paul

In message , dated Mon, 11 Sep 2006, Paul writes

Well, of course. I have two physically identical inductors, one with 50 turns and one with 60. One is here and the other 100 m away. k is very small, but L1/L2 = (N1/N2)^2.

It would be very strange indeed if one inductor at a considerable distance could affect the inductance of another. You could call it 'quantum entrapment' Oh, you have! (;-)

rather

same inductance per

approximately the same

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differ substantially.

thread.

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looking at

transformer is k,

then you can

inductors".

just imagine 2

Instead of saying:

"The N2/N1 ratio is an approximati "The N2/N1 ratio is an approximation which is only acceptably accurate, in all cases of inductor geometry, for closely coupled inductors such as found in ordinary transformers (when they *are* closely coupled, as is the usual case)."

This would allow for the possibility that N2/N1 might equal SQRT(L2/L1) for some particular geometry even if k is low.

Consider it said."

Your comment about 3-phase motors makes me realize that in that situation, the geometries and number of turns in windings would tend to be nearly identical, and the N1:N2:N3 ratios would be nearly the same as L1:L2:l3. You should have brought this (the motor connection) up in your original post and we would have seen the relevance of your point. The thread was about transformers, and I couldn't see what you were getting at. Since SQRT(L1/L2) always works in the case of transformers I couldn't see why bother with N1/N2 since in most of the continuum of possibilities with low k transformers, N1/N2 will be in error.

But I see that for motors the A sub l parameterization could be useful since the windings will fall on just that point in the continuum where N1:N2:N3 give the correct result.

coupling is

I would still like to point out for the interested reader (if anyone is left) that 'close coupling' is only possible when primary and secondary geometry are the same, which is as you point out the case in ordinary transformers. Different coil geometries necessarily result in a lower coupling. If anyone out there wants to build a closely coupled transformer, they should understand that this necessarily implies equal geometry for primary and secondary.

Therefore my preferred formulation would have been: "The N2/N1 ratio is an approximation which is acceptably accurate _only_ in cases of _equal_ inductor geometry, such as necessarily the case for closely coupled inductors such as found in ordinary transformers (when they *are* closely coupled, as is the usual case)."

Another example I have of an unusual transformer is a high voltage (1MV) supply I built 10 years ago. It consisted of a stack of 25 'decks', each generating 40kV. Each deck was supplied from its own oscillating LC circuit, the coil being magnetically coupled with the neighbouring decks, and the whole supplied from 1 coil at the bottom. The thing could be seen as one 25-coils air-cored transformer, each of the coils coupling with all the other coils, k decreasing as distance increased. All the coils were the same (number of turns and geometry), so the L1/L2=N1/N2 approximation was valid although k was 0.7 maximum between direct neighbours.

Paul

that 'close

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understand

Actually this is not absolutely true.

coupled

coupled, as is

While reading this I reminded myself (as Legg reminded us) that if the coils are wound on a high permeability core, the geometries can be quite different and still have high k. Since I have been using an air core pair of coils for the measurements I've quoted in the larger thread, air core has been uppermost on my mind. That's my excuse, and I stand by it. :-)

I took an EC-41 core set and wound two bifilar 14 turn coils in a single layer on the bobbin. I then cut a couple of cardboard discs and put them on top of the single layer, with the two discs very close together. Next I wound 28 turns in a thin disc between the cardboard formers. I then inserted and clamped the two ferrite core halves.

The coupling coefficient between the two 14 turn bifilar coils measured .9995. I then wired the two 14 turn bifilar coils in series aiding, forming a single layer 28 turn winding. The coupling coefficient between this single layer winding and the pancake, or disc, winding measured .9973. It would be hard to get any more different geometries than that, but the two windings are closely coupled nonetheless, due to the (relatively) high permeability core.

I built 10

supplied from

neighbouring

coils coupling